Graphs and Combinatorics

, Volume 27, Issue 3, pp 377–397 | Cite as

(Non)Existence of Pleated Folds: How Paper Folds Between Creases

  • Erik D. DemaineEmail author
  • Martin L. Demaine
  • Vi Hart
  • Gregory N. Price
  • Tomohiro Tachi
Proceedings Paper


We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the model can be folded with additional creases, suggesting that real paper “folds” into this model via small such creases. We conjecture that the circular version of this model, consisting simply of concentric circular creases, also folds without extra creases. At the heart of our results is a new structural theorem characterizing uncreased intrinsically flat surfaces—the portions of paper between the creases. Differential geometry has much to say about the local behavior of such surfaces when they are sufficiently smooth, e.g., that they are torsal ruled. But this classic result is simply false in the context of the whole surface. Our structural characterization tells the whole story, and even applies to surfaces with discontinuities in the second derivative. We use our theorem to prove fundamental properties about how paper folds, for example, that straight creases on the piece of paper must remain piecewise-straight (polygonal) by folding.


Origami Hypar Hyperbolic paraboloid Folding Developable surface 


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Copyright information

© Springer 2011

Authors and Affiliations

  • Erik D. Demaine
    • 1
    Email author
  • Martin L. Demaine
    • 1
  • Vi Hart
    • 2
  • Gregory N. Price
    • 1
  • Tomohiro Tachi
    • 3
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Stony BrookUSA
  3. 3.Graduate School of Arts and SciencesThe University of TokyoTokyoJapan

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